let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,COMPLEX st f is_integrable_on M holds
( Integral (M,f) in COMPLEX & Integral (M,|.f.|) in REAL & |.f.| is_integrable_on M )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,COMPLEX st f is_integrable_on M holds
( Integral (M,f) in COMPLEX & Integral (M,|.f.|) in REAL & |.f.| is_integrable_on M )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,COMPLEX st f is_integrable_on M holds
( Integral (M,f) in COMPLEX & Integral (M,|.f.|) in REAL & |.f.| is_integrable_on M )
let f be PartFunc of X,COMPLEX; :: thesis: ( f is_integrable_on M implies ( Integral (M,f) in COMPLEX & Integral (M,|.f.|) in REAL & |.f.| is_integrable_on M ) )
assume A1: f is_integrable_on M ; :: thesis: ( Integral (M,f) in COMPLEX & Integral (M,|.f.|) in REAL & |.f.| is_integrable_on M )
reconsider AF = |.f.| as PartFunc of X,REAL ;
AF is_integrable_on M by A1, MESFUN7C:35;
hence ( Integral (M,f) in COMPLEX & Integral (M,|.f.|) in REAL & |.f.| is_integrable_on M ) by LPSPACE1:44, XCMPLX_0:def 2; :: thesis: verum